A038338 → basis A013609 → D+1

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Submission text ?

Triangle begins:
  0:              1728,               2304,                576;
  1:             55296,             124416,              82944,               13824;
  2:           1658880,            5308416,            5971968,             2654208,              331776;
  3:          47775744,          199065600,          318504960,           238878720,            79626240,             7962624;
  4:        1337720832,         6879707136,        14332723200,         15288238080,          8599633920,          2293235712,           191102976;
  5:       36691771392,       224737099776,       577895399424,        802632499200,        642105999360,        288947699712,         64210599936,          4586471424;
  6:      990677827584,      7044820107264,     21574761578496,      36985305563136,      38526359961600,      24656870375424,       9246326390784,       1761205026816,        110075314176;
  7:    26418075402240,    213986410758144,    760840571584512,    1553382833651712,    1997206500409344,    1664338750341120,     887647333515264,     285315214344192,      47552535724032,      2641807540224;
  8:   697437190619136,   6340338096537600,  25678369290977280,   60867245726760960,   93202970019102720,   95865912019648512,   66573550013644800,   30433622863380480,    8559456430325760,   1268067619307520,    63403380965376;
  9: 18260173718028288, 184123418323451904, 836924628742963200, 2259696497606000640, 4017238217966223360, 4921116817008623616, 4218100128864534528, 2510773886228889600, 1004309554491555840, 251077388622888960, 33476985149718528, 1521681143169024;

The row polynomials satisfy (n+2+x)*P(n,x) = (24*n+72+(24*n+96)*x+24*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1728 + 2304*x + 576*x^2.

P[0, x_] := 1728 + 2304*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24*n + 72 + (24*n + 96)*x + 24*x^2)*P[n - 1, x])/(n + 2 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1728 + 2304*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24*n + 72 + (24*n + 96)*x + 24*x^2)*v[i - 1])/(n + 2 + x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Laguerre’s theorem on differential operators
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Laguerre’s theorem on differential operators
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from Laguerre’s theorem on differential operators
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Laguerre’s theorem on differential operators

Proof chains ? JSON

• Laguerre’s theorem on differential operators [SP14] polytool ? step ›

  • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
    • Geometric basis map (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*4^j.?
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

      • from basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
        • ↑ Geometric basis map (elementary) (above)
        • Geometric basis map (elementary) polytool ? step ›

          • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
            • Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check
            • Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
            • Parent-row provers of polytool polytool ? step › direct check
            • Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
            • Affine, reversal and Möbius substitutions (elementary) polytool ? step ›

              • from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*12^j.?
                • ↑ Row-hypergeometric rows (multiplier sequences) (above)
                • ↑ Liu–Wang theorem (above)
                • ↑ Parent-row provers of polytool (above)
                • ↑ Iterated real-rootedness preservers (above)
    • ↑ Geometric basis map (elementary) (above)

Related ?

Same fingerprint ?

• A135838 → D+1
• A120910 → basis A013610 → D+1
• A376827 → θ+1
• A085881 → basis A013609 → θ+1
• A094305 → θ+1
• A178822 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → D+1
• A093375 → D
• A038330 → basis A013611 → D+1
• A038338 → D+1
• basis-map coefficients of A113214, shifted → D → D+1
• basis-map coefficients of A113214, shifted → θ+1
• A007318 → ×(−(1/2)n+(1/2)n²) → D+1
• A093375 → D → D
• A007318 → ×(−(1/2)n+(1/2)n²) → θ+1
• A085880 → θ+1
• A356546 → θ+1
• A216973 → θ+1
• A085881 → D+1
• A028326 → D+1
• A178820 → basis A013609 → θ+1
• A134346 → θ+1
• A028326 → D → D+1
• A038208 → D → D+1
• basis-map coefficients of A026729, shifted → basis A007318 → D+1
• A135065 → D+1
• A038330 → basis A013611 → θ+1
• A135838 → θ+1
• A124860 → D+1
• A038338 → basis A013609 → θ+1
• A038325 → θ+1
• A085880 → D+1
• A038221 → basis A013609 → D+1
• A038221 → basis A013609 → θ+1
• A124860 → θ+1
• A356546 → basis A013609 → θ+1
• basis-map coefficients of A112227 → x ↦ x+1 → θ+1
• basis-map coefficients of A026729 → basis A007318 → θ+1
• A028326 → basis A013609 → θ+1
• basis-map coefficients of A026729 → basis A007318 → D+1
• A038221 → D+1
• A085881 → θ+1
• A038325 → basis A013609 → θ+1
• A356546 → basis A013609 → D+1
• basis-map coefficients of A026729, shifted → basis A007318 → θ+1
• A038208 → D → θ+1
• A134346 → D+1
• A196347 → θ+1
• A196347 → basis A013609 → D+1
• A156992 → basis A013609 → D+1
• A003506 → D → θ+1
• A178822 → basis A013609 → θ+1
• A038208 → D+1
• A120910 → basis A013610 → θ+1
• A178822 → basis A013609 → D+1
• basis-map coefficients of A127647 → θ+1
• A376827 → D+1
• A085880 → D → θ+1
• A085880 → basis A013609 → θ+1
• A131431 → D+1 → basis A007318
• A178821 → basis A013609 → D+1
• A131431 → x ↦ x+1 → θ+1
• A028326 → θ+1
• A038208 → θ+1
• A178820 → D+1
• A028326 → D → θ+1
• A156992 → D+1
• A178821 → basis A013609 → θ+1
• A093375: Array T(m,n) read by ascending antidiagonals: T(m,n) = m*binomial(n+m-2, n-1) for m, n >= 1.
• A356546 → D+1
• A038325 → basis A013609 → D+1
• A085880 → D → D+1
• A344565 → θ+1
• A344565 → D+1
• A178822 → D+1
• A196347 → basis A013609 → θ+1
• basis-map coefficients of A113214, shifted → D → θ+1
• A131431 → x ↦ x+1 → D+1
• A196347 → D+1
• A156992 → basis A013609 → θ+1
• A178821 → D+1
• basis-map coefficients of A113214, shifted → D+1
• A038221 → θ+1
• A178820 → basis A013609 → D+1
• A085880 → basis A013609 → D+1
• A178820 → θ+1
• basis-map coefficients of A113214, shifted → basis A013609 → D+1
• basis-map coefficients of A127647 → D+1
• A156992 → θ+1
• A027465 → basis A013611 → θ+1
• A027465 → basis A013611 → D+1
• A038338 → θ+1
• A085881 → basis A013609 → D+1
• A038325 → D+1
• A178821 → θ+1
• A103406: Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix w…
• A135065 → θ+1
• A028326 → basis A013609 → D+1
• A094305 → D+1
• basis-map coefficients of A113214, shifted → basis A013609 → θ+1

References

1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.